Multiple-Frequency imaging

Figure 75-2 shows third-order data or a hyperspectral data cube where the spectral amplitude is measured at multiple frequencies (spectrum) with X and Y spatial dimensions included. Each plane in the figure represents the amplitude of the spectral signal at a single frequency for an X and Y coordinate spatial image. 

All the 2-D discrete transforms mentioned above analyze the data at one scale or resolution only. Over the last 20 years, new transforms that split the image into multiple frequency bands have given rise to schemes that analyze images at multiple scales. These transforms, known as wavelet transforms, have long been known to mathematicians. The implementation of wavelet transforms using filter banks has enabled the development of many new transforms. 

Fig. 1.4a,b. Multiple-frequency transducers. a,b Axial images of the penile shaft obtained with an 8-5 MHz multiple frequency transducer (15L8, Sequoia, Siemens) by setting the center frequency at a 8 MHz and b 14 MHz, respectively. Shifting the frequency band upward, a more defined echotex-ture is appreciated as a result of an increased resolution... 

Fig. 1.21 Echo Planar Imaging (EPI) pulse sequence. Gradient-echo based multiple echoes are used for fast single-shot 2D imaging. Slice selection along Gs and frequency encoding along C, are utilized. Phase encoding is realized using short blipped gradient pulses along Gp.

Another approach to obtain spatially selective chemical shift information is, instead of obtaining the entire image, to select only the voxel of interest of the sample and record a spectrum. This method called Volume Selective spectroscopY (VOSY) is a ID NMR method and is accordingly fast compared with a 3D sequence such as the CSI method displayed in Figure 1.25(a). In Figure 1.25(b), a VOSY sequence based on a stimulated echo sequence is displayed, where three slice selective pulses excite coherences only inside the voxel of interest. The offset frequency of the slice selective pulse defines the location of the voxel. Along the receiver axis (rx) all echoes created by a stimulated echo sequence are displayed. The echoes V2, VI, L2 and L3 can be utilized, where such multiple echoes can be employed for signal accumulation. 

Fig. 3.8. A one-dimensional spatial frequency diagram. A spatial frequency component with periodicity ajq is represented by the point with coordinate Kt - In/ajq on the 1C-axis. If a digital image recording system has a pixel spacing of ap, then periodic structures whose spatial frequency lies outside the range —n/ap < K < +n/ap will appear as structures with spatial frequency shifted by an integral multiple of 2n/ap to...

What happens if you try to image an object with periodicity outside this spatial frequency range The spatial frequency will be displaced by an integral multiple of 2n/a so as to lie in the range that the framestore can handle. There is an almost exact analogy with phonon wavevectors in a crystal lattice or, if you prefer, with why stage-coach wheels appear to go backwards in movies. The effect is illustrated in Fig. 3.8 with a feature with a periodic structure of spatial frequency Kj. It will be stored in the framestore with a spatial frequency dz JCj — 27T K/ap, with ng = 1 in the example here. This will appear as a periodic structure in the image that bears no apparent relationship to the object. 

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